class: center, middle, inverse, title-slide .title[ # Class 2b: Review of concepts in Probability and Statistics ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> --- layout: false class: inverse, middle # Summarizing Data ## Summary Statistics --- # Measures of Central Tendency ## Mean - **Mean** represents the arithmetic average of the data. - Sometimes called the expected value of the random variable E(X) - The population mean `\(\mu\)` is the sum of all observations divided by the total population size: `$$\mu =E(X)=\frac{\sum_{i=1}^{N} x_i}{N}=\sum_{x\in X}P(X=x) \times x$$` - where `\(N\)` is the total population size, and `\(x_i\)` are individual data points. - The sample mean, denoted as `\(\bar{x}\)`, is the sample equivalent: `$$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1+x_2+...x_{n-1}+x_n}{n}$$` where `\(n\)` is the sample size. --- ## Mean Intuitively, mean is the balancing point of the distribution. <!-- --> --- ## Mean of a binary variable What if a mean of a **binary variable**? - Binary variable is a variable which takes value 0 or 1 - For example: do you have diabetes (yes=1, no=0) -- What is the intuitive interpretation of the mean of this variable? - `\(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)` - `\(\bar{x} = \frac{1+0+0+...0+1}{n}=\frac{n_{diabetes}}{n}=\hat{\mu}_{diabetes}\)` -- It's the proportion of people with diabetes in the sample: mean(diabetes)= 0.11 --- ## Weighted Mean - In some scenarios, data points have different weights. - For a dataset with weights `\(w_i\)` and values `\(x_i\)`, the weighted mean is: `$$\small \text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i \cdot x_i}{\sum_{i=1}^{n} w_i}$$` <div class="datatables html-widget html-fill-item" id="htmlwidget-b7613d44a90b027911f1" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-b7613d44a90b027911f1">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Midterm 1","Midterm 2","Quizzes","Final Project","Final exam"],[0.2,0.2,0.15,0.15,0.3],[6,8,9,4,8]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Person<\/th>\n <th>Weight<\/th>\n <th>Grade<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"Person","targets":0},{"name":"Weight","targets":1},{"name":"Grade","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> The ** weighted mean** is: `\begin{align*} \small \bar{x} & =\frac{0.2\times 6+0.2\times 8+0.15 \times 9+ 0.15 \times 4+0.3 \times 8}{0.2+0.2+0.15+0.15+0.3} \end{align*}` --- ## Mean - Is mean always a right measure? #### "Bill Gates walks into a bar" - Suppose a group of people, including Bill Gates, walks into a bar. - Let's say the net worth of everyone in the group is as follows: .pull-left[ <div class="datatables html-widget html-fill-item" id="htmlwidget-b1aec08772e71d9332a1" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-b1aec08772e71d9332a1">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Person 1","Person 2","Person 3","Person 4","Person 5","Bill Gates"],[10,20,30,40,50,600000]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Person<\/th>\n <th>Net_Worth<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":1},{"name":"Person","targets":0},{"name":"Net_Worth","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] .pull-right[ The **mean** is: `\begin{align*} \bar{x} & =\frac{10 + 20 + 30 + 40 + 50 + 60000}{6} \\ & = 100025 \\ \end{align*}` Mean is seriously skewed due to the outlier. ] --- ## Mean vs Median <center> <img src=mean_median.jpg width="800"> </center> --- ## Median - **Median** represents the middle value when data is sorted - Half of observations are below it, half are above it. - For a dataset with odd size `\(n\)`, the median is the `\(\frac{n+1}{2}\)`-th value - For even size `\(n\)`, it's the average of `\(\frac{n}{2}\)`-th and `\(\frac{n}{2}+1\)`-th values. .pull-left[ | Day | Number of Customers | |-----|---------------------| | 1 | 20 | | 2 | 18 | | 3 | 25 | | 4 | 22 | | 5 | 30 | | 6 | 21 | | 7 | 27 | ] .pull-right[ The dataset has `\(n=7\)` (odd) observations, so to find the median: - Arrange the data in ascending order: - 18, 20, 21, 22, 25, 27, 30. - The median is the `\(\frac{n+1}{2}\)`-th value, which is the 4th value. - Thus, the median is the 4th value, which is 22. ] --- ### Let's look at the median weight in our population - Mean: 72.66451 - Median: 70.7536 -- <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/unnamed-chunk-5-1.png" width="100%" /> - Mean is dotted - Median is dashed --- ### Median and outliers I added couple of observations on the right tail of the distribution - Old Mean: 72.66, **New Mean: 77.05** - Old Median: 70.75, **New Median: 70.95** <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/abc-1.png" width="100%" /> --- ## Side note on the Mode **Mode** is the most frequent value in the data - Let's look at the distribution of age of people with diabtese <div class="datatables html-widget html-fill-item" id="htmlwidget-bf14b3fe527cf9eaee51" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-bf14b3fe527cf9eaee51">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97"],[4,2,4,3,5,7,6,6,5,12,16,14,16,15,33,33,31,32,48,61,37,47,82,73,65,75,67,85,64,114,98,87,115,99,105,103,124,100,122,119,133,99,114,115,107,129,112,112,116,96,90,74,82,91,69,73,59,45,51,46,45,17,41,23,25,18,13,13,10,5,4,1,4,1,2,1,1,1],[0.001,0,0.001,0.001,0.001,0.002,0.001,0.001,0.001,0.003,0.004,0.003,0.004,0.004,0.008,0.008,0.007,0.008,0.012,0.015,0.008999999999999999,0.011,0.02,0.017,0.016,0.018,0.016,0.02,0.015,0.027,0.023,0.021,0.028,0.024,0.025,0.025,0.03,0.024,0.029,0.029,0.032,0.024,0.027,0.028,0.026,0.031,0.027,0.027,0.028,0.023,0.022,0.018,0.02,0.022,0.017,0.017,0.014,0.011,0.012,0.011,0.011,0.004,0.01,0.006,0.006,0.004,0.003,0.003,0.002,0.001,0.001,0,0.001,0,0,0,0,0]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Age<\/th>\n <th>n_i<\/th>\n <th>p_i<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"Age","targets":0},{"name":"n_i","targets":1},{"name":"p_i","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ## Mode <center> <img src=mode.jpg width="400"> </center> --- ## Percentiles <center> <img src=Natalie.jpg width="300"> </center> --- ## Percentiles <center> <img src=t_swift2.jpg width="300"> </center> --- ## Percentiles <center> <img src=t_swift_01.jpg width="300"> </center> --- --- ## Percentiles <center> <img src=t_swift_0001.jpg width="300"> </center> --- ## Percentiles <center> <img src=WSJ.jpg width="800"> </center> Credits: Wall Street Journal [Article Link](https://www.wsj.com/tech/personal-tech/spotify-wrapped-2023-taylor-swift-e303333d) --- ## Percentiles - How much inventory of milk you need to keep in your Starbucks? -- - What is the tradeoff of keeping too much vs too litle inventory? -- - Suppose we want to have enough of milk to cover sales on 95% of days -- - To figure it out, let's look at the distribution of the daily use of milk <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/Sales_dist_figure-1.png" width="100%" /> --- ## Percentiles - Let `\(s_i\)` be the daily sales of milk - We want to choose amount `\(M\)`, such that `\(P(s_i \leq M)=0.95\)` - That is, in 95% of days sales are smaller or equal than `\(M\)` -- <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/Sales_dist_figure_with_shaded_region-1.png" width="100%" /> -- - What is this number? - It's the 95th percentile of the distribution (274 liters) --- ## Percentiles - *Percentiles* divide the ordered data into 100 equal parts. - `\(p\)`th percentile is a value such that `\(p\%\)` of the data are below it - `\(v_p\)` is such that `\(P(x_i \leq v_p)=p\)` - `\(v_{95}\)` is such that `\(P(x_i \leq v_{95})=95\%\)` --- ## Percentiles -- - What is the the height such that 75% of ITAM students are smaller than this height? -- - What is the income level such that 25% of people in Mexico earn less than that level? -- - What is the age, such that 50% of people die before that age? --- ## Percentiles <center> <img src=Exam_q_top5.png width="800"> </center> --- ## How to find it in a sample 1. Arrange the data in ascending order -- 2. Find which observation corresponds to the relevant percentile - Formula: `\(i = \left(\frac{p}{100}\right)(n+1)\)` - Example: To find 95th percentile in a sample of 1000 observations we look at `\(i = \left(\frac{95}{100}\right)(1000+1)=950.95\)` observation -- 3. If it's an integer, value of ith observation is your percentile 4. If it's not, take the average between ith rounded down and ith rounded up - In our example it would be the average of 950th and 951th observation --- ## Or use the CDF - `\(ECDF(v)=P(x_i \leq v)\)` <div id="htmlwidget-2717a99ca11583d690eb" style="width:800px;height:300px;" class="plotly html-widget"></div> <script type="application/json" 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income is below the 25th percentile <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/Image_with_shaded_area_Qtiles-1.png" width="100%" /> --- - **Deciles** - 10th, 20th, ... 90th - How bad pollution gets in CDMX during top 10% polluted days? - During top 10% of polluted days pollution level is larger or than 9th decile. <img src="data:image/png;base64,#C_2_slides_b_files/figure-html/Image_with_shaded_area_Deciles-1.png" width="100%" /> --- ### Example with data Here is a data on distribution of how many views have various tik-tok videos. - What is the 1st decile? - What is the 95th percentile? <div class="datatables html-widget html-fill-item" id="htmlwidget-7693b226083ebd1214ea" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7693b226083ebd1214ea">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["TikTok Video 1","TikTok Video 2","TikTok Video 3","TikTok Video 4","TikTok Video 5","TikTok Video 6","TikTok Video 7","TikTok Video 8","TikTok Video 9","TikTok Video 10","TikTok Video 11","TikTok Video 12","TikTok Video 13","TikTok Video 14","TikTok Video 15","TikTok Video 16","TikTok Video 17","TikTok Video 18","TikTok Video 19","TikTok Video 20","TikTok Video 21","TikTok Video 22","TikTok Video 23","TikTok Video 24","TikTok Video 25","TikTok Video 26","TikTok Video 27","TikTok Video 28","TikTok Video 29","TikTok Video 30","TikTok Video 31","TikTok Video 32","TikTok Video 33","TikTok Video 34","TikTok Video 35","TikTok Video 36","TikTok Video 37","TikTok Video 38","TikTok Video 39","TikTok Video 40","TikTok Video 41","TikTok Video 42","TikTok Video 43","TikTok Video 44","TikTok Video 45","TikTok Video 46","TikTok Video 47","TikTok Video 48","TikTok Video 49","TikTok Video 50","TikTok Video 51","TikTok Video 52","TikTok Video 53","TikTok Video 54","TikTok Video 55","TikTok Video 56","TikTok Video 57","TikTok Video 58","TikTok Video 59","TikTok Video 60","TikTok Video 61","TikTok Video 62","TikTok Video 63","TikTok Video 64","TikTok Video 65","TikTok Video 66","TikTok Video 67","TikTok Video 68","TikTok Video 69","TikTok Video 70","TikTok Video 71","TikTok Video 72","TikTok Video 73","TikTok Video 74","TikTok Video 75","TikTok Video 76","TikTok Video 77","TikTok Video 78","TikTok Video 79","TikTok Video 80","TikTok Video 81","TikTok Video 82","TikTok Video 83","TikTok Video 84","TikTok Video 85","TikTok Video 86","TikTok Video 87","TikTok Video 88","TikTok Video 89","TikTok Video 90","TikTok Video 91","TikTok Video 92","TikTok Video 93","TikTok Video 94","TikTok Video 95","TikTok Video 96","TikTok Video 97","TikTok Video 98","TikTok Video 99","TikTok Video 100","TikTok Video 101","TikTok Video 102","TikTok Video 103","TikTok Video 104","TikTok Video 105","TikTok Video 106","TikTok Video 107","TikTok Video 108","TikTok Video 109","TikTok Video 110","TikTok Video 111","TikTok Video 112","TikTok Video 113","TikTok Video 114","TikTok Video 115","TikTok Video 116","TikTok Video 117","TikTok Video 118","TikTok Video 119","TikTok Video 120","TikTok Video 121","TikTok Video 122","TikTok Video 123","TikTok Video 124","TikTok Video 125","TikTok Video 126","TikTok Video 127","TikTok Video 128","TikTok Video 129","TikTok Video 130","TikTok Video 131","TikTok Video 132","TikTok Video 133","TikTok Video 134","TikTok Video 135","TikTok Video 136","TikTok Video 137","TikTok Video 138","TikTok Video 139","TikTok Video 140","TikTok Video 141","TikTok Video 142","TikTok Video 143","TikTok Video 144","TikTok Video 145","TikTok Video 146","TikTok Video 147","TikTok Video 148","TikTok Video 149","TikTok Video 150","TikTok Video 151","TikTok Video 152","TikTok Video 153","TikTok Video 154","TikTok Video 155","TikTok Video 156","TikTok Video 157","TikTok Video 158","TikTok Video 159","TikTok Video 160","TikTok Video 161","TikTok Video 162","TikTok Video 163","TikTok Video 164","TikTok Video 165","TikTok Video 166","TikTok Video 167","TikTok Video 168","TikTok Video 169","TikTok Video 170","TikTok Video 171","TikTok Video 172","TikTok Video 173","TikTok Video 174","TikTok Video 175","TikTok Video 176","TikTok Video 177","TikTok Video 178","TikTok Video 179","TikTok Video 180","TikTok Video 181","TikTok Video 182","TikTok Video 183","TikTok Video 184","TikTok Video 185","TikTok Video 186","TikTok Video 187","TikTok Video 188","TikTok Video 189","TikTok Video 190","TikTok Video 191","TikTok Video 192","TikTok Video 193","TikTok Video 194","TikTok Video 195","TikTok Video 196","TikTok Video 197","TikTok Video 198","TikTok Video 199","TikTok Video 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class=\"display\">\n <thead>\n <tr>\n <th>VideoTitle<\/th>\n <th>Views<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":1},{"name":"VideoTitle","targets":0},{"name":"Views","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- - Index for the first decile is: `\(i = \left(\frac{10}{100}\right)(200+1)=20.1\)` - First decile is the average of the 20th and 21st observation - Index for the 95th percentile is: `\(i = \left(\frac{95}{100}\right)(200+1)=190.95\)` - 95th percentile is the average of the at 190th and 191st observation --- ### Exercises: - Review Exercises: - PDF 2: 3,4,5,6, - Homeworks - Lista 00.1: 3